Express as a single logarithm
ln(a)+1/2ln(b)
2ln4-ln2
To express the expression ln(a) + 1/2 ln(b) as a single logarithm, we can make use of the logarithmic properties.
First, let's simplify ln(a) + 1/2 ln(b):
ln(a) + 1/2 ln(b) = ln(a) + ln(b^(1/2))
Using the property ln(x) + ln(y) = ln(x * y), we can combine the logarithms:
ln(a) + ln(b^(1/2)) = ln(a * b^(1/2))
To simplify further, we can use the property ln(x^y) = y * ln(x):
ln(a * b^(1/2)) = ln(a) + ln(b^(1/2)) = ln(a) + (1/2) ln(b) = ln(a) + ln(b^(1/2)) = ln(a * b^(1/2))
Therefore, the expression ln(a) + 1/2 ln(b) can be expressed as ln(a * b^(1/2)).
Now, let's simplify the expression 2 ln(4) - ln(2):
Using the property ln(x^n) = n ln(x), we can simplify:
2 ln(4) - ln(2) = ln(4^2) - ln(2)
Simplifying further using the property ln(x/y) = ln(x) - ln(y):
ln(4^2) - ln(2) = ln(16) - ln(2)
Finally, we can use the property ln(x * y) = ln(x) + ln(y) to combine the logarithms:
ln(16) - ln(2) = ln(16/2)
Therefore, the expression 2 ln(4) - ln(2) can be expressed as ln(16/2).